Multiple choice

$A$ flag staff stands upon the top of a building. $A$t a distance of 40 $m$. the angles of elevation of the tops of the flag staff and building are $60^{ }$ and $30^{0}$ then the height of the flag staff in metres is

  1. $40\sqrt{3}$
  2. $\displaystyle \frac{40}{\sqrt{3}}$
  3. $\displaystyle \frac{160}{\sqrt{3}}$
  4. $\displaystyle \frac{80}{\sqrt{3}}$
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D Correct answer
Explanation

Let h be the height of the building and H be the height of the flag staff. The height of the top of the building is 40 * tan(30) = 40/sqrt(3). The height of the top of the flag staff is 40 * tan(60) = 40 * sqrt(3). The flag staff height is 40 * sqrt(3) - 40/sqrt(3) = (120 - 40) / sqrt(3) = 80/sqrt(3).

AI explanation

Let the height of the building be h and the height of the flag staff be f, so the total height to the top of the flag staff is h plus f. Using the tangent ratio for the building, tan(30) = h / 40, so h = 40 / sqrt(3). For the flag staff, tan(60) = (h + f) / 40, and since tan(60) is sqrt(3), we have h plus f equals 40 * sqrt(3). Subtracting the building height from the total height gives f = 40 * sqrt(3) - 40 / sqrt(3), which simplifies to 80 / sqrt(3). The result is 80 / sqrt(3).

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